Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance ( R / 2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :

Question

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance ( R / 2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period : (A) (2 π R/g) (B) ( g /2 π R ) (C) (1/2 π) √( g / R ) (D) 2 π √( R / g )

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/physics/000237e20e0b50ea438729cddf0a25eb-.png" /> Force along the tunnel F =-(( GMmr / R 3)) cos θ F =-( gm / R ) x (( GM / R 2)= g , r cos θ= x ) a=-(g/R) x ω2=( g / R ) T =2 π √( R / g )

$\frac{2 π R}{g}$

$\frac{g}{2 π R}$

$\frac{1}{2 π} \frac{g}{R}$

$2 π \frac{R}{g}$

Force along the tunnel
$F = - (\frac{GM m r}{R ^{3}}) cos θ$
$F = - \frac{g m}{R} x (\frac{GM}{R ^{2}} = g, r cos θ = x)$
$a = - \frac{g}{R} x$
$ω^{2} = \frac{g}{R} T = 2 π \frac{R}{g}$

Q. Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (R/2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :

g2πR​

2πRg​

2π1​Rg​​

2πgR​​

Force along the tunnel F=−(R3GMmr​)cosθ F=−Rgm​x(R2GM​=g,rcosθ=x) a=−Rg​x ω2=Rg​T=2πgR​​

$\frac{2 π R}{g}$

$\frac{g}{2 π R}$

$\frac{1}{2 π} \frac{g}{R}$

$2 π \frac{R}{g}$

Force along the tunnel
$F = - (\frac{GM m r}{R ^{3}}) cos θ$
$F = - \frac{g m}{R} x (\frac{GM}{R ^{2}} = g, r cos θ = x)$
$a = - \frac{g}{R} x$
$ω^{2} = \frac{g}{R} T = 2 π \frac{R}{g}$